Ideals and idempotents in the uniform ultrafilters

TL;DR

This paper investigates the structure of uniform ultrafilters on discrete semigroups, establishing conditions under which they form ideals containing minimal left ideals and idempotents, with implications for countable semigroups.

Contribution

The paper provides general conditions for uniform ultrafilters to contain prime minimal left ideals and maximal idempotents, including specific results for countable, weakly cancellative semigroups.

Findings

U(S) can be a two-sided ideal under mild conditions

Countable weakly cancellative semigroups have prime minimal left ideals in U(S)

Examples show the sharpness of the conditions

Abstract

If $S$ is a discrete semigroup, then $\beta S$ has a natural, left-topological semigroup structure extending $S$. Under some very mild conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided ideal of $\beta S$, and therefore contains all of its minimal left ideals and minimal idempotents. We find some very general conditions under which $U(S)$ contains prime minimal left ideals and left-maximal idempotents. If $S$ is countable, then $U(S) = S^*$, and a special case of our main theorem is that if a countable discrete semigroup $S$ is a weakly cancellative and left-cancellative, then $S^*$ contains prime minimal left ideals and left-maximal idempotents. We will provide examples of weakly cancellative semigroups where these conclusions fail, thus showing that this result is sharp.